Ohm's law and resistance
Resistance relates voltage and current in the simplest and most common circuit element. Ohm's law is the first tool of every circuit calculation.
Ohm's law
Georg Ohm published his law in 1827: for a resistor, voltage is proportional to current, \(v = iR\), where R is the resistance in ohms (\(\Omega\)). Equivalently \(i = \frac{v}{R}\) and \(R = \frac{v}{i}\). A 1 k\(\Omega\) resistor with 5 V across it carries 5 mA. Materials and devices that obey this proportionality are called ohmic.
Resistivity
A conductor's resistance depends on its material and shape: \(R = \frac{\rho L}{A}\), where \(\rho\) is resistivity, L length, and A cross-sectional area. Copper's resistivity is about \(1.7 \times 10^{-8}\) \(\Omega\)m, which is why it is used for wiring; insulators such as glass have resistivities more than \(10^{20}\) times larger. Longer, thinner wires have more resistance.
Conductance
Conductance is the reciprocal of resistance, G = 1/R, measured in siemens (S). It is convenient when analysing parallel circuits, where conductances add. A short circuit has zero resistance (infinite conductance); an open circuit has infinite resistance (zero conductance), so no current flows however large the voltage.
Power in resistors
Combining \(p = vi\) with Ohm's law gives three forms: \(p = i^2 R = \frac{v^2}{R}\). Resistors convert electrical energy into heat, which is useful in heaters and toasters but a loss elsewhere. Real resistors have power ratings, such as 0.25 W; exceeding them overheats and damages the part.
Real resistors
Practical resistors are made of carbon film, metal film, wire wound on a core, or thin films on chips. They come in standard series of values, such as the E12 series (10, 12, 15, 18, 22, 27, 33, 39, 47, 56, 68, 82 times powers of ten), with tolerances such as 5% or 1%. Through-hole resistors use coloured bands to mark value; surface-mount parts use numeric codes.
Temperature and non-ohmic devices
Resistance changes with temperature: metals' resistance rises as they heat, which is used in platinum resistance thermometers, while thermistors change sharply with temperature. Many devices are non-ohmic: a filament lamp's resistance increases greatly as it heats, and diodes conduct easily in one direction only. Ohm's law applies to resistors, not to everything.
Measurement
A multimeter measures voltage (connected in parallel across the element), current (connected in series, breaking the circuit to insert the meter), and resistance (with the circuit unpowered). An ideal voltmeter draws no current, having infinite resistance, and an ideal ammeter has zero resistance; real meters approximate these, which matters in sensitive circuits.
Worked example
A 1.2 kW electric kettle runs on 230 V mains. Its current is \(i = \frac{p}{v} = \frac{1200}{230} \approx 5.2\) A, and its element's resistance is \(R = \frac{v^2}{p} = \frac{230^2}{1200} \approx 44\) \(\Omega\). Boiling for 3 minutes uses \(1.2 \times 0.05 = 0.06\) kWh, costing about 1.8 cents at 30 cents per kWh.
Key takeaways
Ohm's law (1827) states v = iR for resistors, in ohms. Resistance depends on resistivity, length, and area: R = rho L / A; copper has low resistivity. Conductance G = 1/R is in siemens; shorts have zero resistance and opens infinite. Resistor power is p = i^2 R = v^2 / R, turned into heat, limited by power ratings. Real resistors come in standard series such as E12 with tolerances. Resistance varies with temperature, and lamps and diodes are non-ohmic. Voltmeters connect in parallel, ammeters in series, ohmmeters on unpowered circuits.
Flashcard
v = iR
Check your understanding
A 100 ohm resistor has 0.2 A flowing through it. How much power does it dissipate?